LESSON 2 • Exponential Decay 335

IInvest invest iggationation 55 Square Roots and Radicals Square Roots and RadicalsIn your work on problems of insulin decay, you found that some questions required calculation with exponential expressions involving a fractional base and fractional powers. For example, estimating the amount of insulin active in the bloodstream 1.5 minutes after a 10-unit injection required calculating 10(0.951.5).

Among the most useful expressions with fractional exponents are those with power one-half. It turns out that one-half powers are connected to the square roots that are so useful in geometric calculations like those involving the Pythagorean Theorem. For any non-negative number b,

b 1 _ 2 = √ " b .

Expressions like √ " b , √ " 5 , and √ """ 9 - x2 are called radicals. As you work on

the following problems, keep this question in mind:

How can you use your understanding of properties of exponents to guide your thinking about one-half powers, square roots, radical

expressions, and rules for operating with them?

1 For integer exponents m and n, you know that (am)n = amn. That property can be extended to work with fractional exponents.

a. Write each of these expressions in standard number form without exponents or radicals.

i. ( 2 1 _ 2 ) 2

ii. ( 5 1 _ 2 ) 2 iii. ( 12

1 _ 2 ) 2 iv. ( 2.4

1 _ 2 ) 2

b. How do the results of Part a explain why the definition b 1 _ 2 = √ " b

makes sense?

2 Write each of the following expressions in an equivalent form using radicals and then in simplest number form (without exponents or radicals).

a. (25) 1 _ 2 b. (9)

1 _ 2 c. ( 9 _ 4 )

1 _ 2 d. (100)

1 _ 2

3 The diagram below shows a series of squares with side lengths increasing in sequence 1, 2, 3, 4, and one diagonal drawn in each square.

a. Use the Pythagorean Theorem to find the exact length of the diagonal of each square.

b. How are the lengths of the diagonals in the three larger squares related to the length of the diagonal of the unit square?

Name: ____________________________________

Unit 3Lesson 2

Investigation 5

336 UNIT 5 • Exponential Functions

c. Look for a pattern in the results of Part b to complete the statement beginning:

The length d of each diagonal in a square with sides of length s is given by d = ... .

4 The pattern relating side and diagonal lengths in a square illustrates a useful rule for simplifying radical expressions:

For any non-negative numbers a and b: √ " ab =

√ " a √ " b .

a. What properties of square roots and exponents justify the steps in this argument? For any non-negative numbers a and b:

√ " ab = (ab )

1 _ 2 (1)

= a 1 _ 2 b

1 _ 2 (2)

= √ " a

√ " b (3)

b. Modify the argument in Part a to justify this property of radicals:

For any non-negative numbers a and b (b ≠ 0),

√ " a _ b =

√ " a _

√ " b

.

5 Use the properties of square roots in Problem 4 to write expressions a–h in several equivalent forms. In each case, try to find the simplest equivalent form—one that involves only one radical and the smallest possible number inside that radical. Check your ideas with calculator estimates of each form. For example,

√ " 48 =

√ " 4 √ " 12

= 2 √ " 12

= 2 √ " 4

√ " 3

= 2 · 2 √ " 3

= 4 √ " 3

Calculator estimates show that √ " 48 ≈ 6.93 and 4

√ " 3 ≈ 6.93.

a. √ "" 9 · 5 b.

√ " 18 √ " 8 c.

√ " 45 d. √ "" 4 · 9

e.

√ "" 4 · 1 _ 9 f.

√ " 9 _ 4 g. √ " 12 h.

√ " 96

6 The properties of square roots in Problem 4 are like distributive properties—taking the square root distributes over the product or the quotient of two (or more) numbers. One of the most common errors in working with square roots is distributing the square root sign over addition. However,

√ """ a + b ≠

√ " a + √ " b

except in some very special cases. Use several pairs of positive values for a and b to show that taking square roots does not distribute over addition (or subtraction).

LESSON 2 • Exponential Decay 337

Check Your UnderstandingCheck Your UnderstandingUse your understanding of fractional exponents and radical expressions to help complete the following tasks.

a. How could you use a calculator with only +, -, ×, and ÷ keys to check these claims about values of expressions involving fractional exponents?

i. 22 5 1 _ 2 = 15 ii. 7

1 _ 2 ≈ 2.65

b. Find the values of these expressions, without use of a calculator.

i. 3 6 1 _ 2 ii.

√ $ 81

iii. ( 25 _ 16 )

1 _ 2 iv.

√ $$ 49

_ 81

c. Use the property that for non-negative numbers a and b, √ $ ab = √ $ a √ $ b to help write each of these radical expressions in at least two equivalent forms.

i. √ $ 30 ii. √ $ 10

√ $ 40

iii. √ $$ 7 _ 25 iv. √ $ 2 _ 3

√ $ 3 _ 2

d. What is the length of each diagonal in a square with sides of length 7 centimeters?

e. Give a counterexample to show that for nonnegative numbers a and b,

√ $$$ a - b is not equal to √ $ a -

√ $ b

Summarize the Mathematics

In this investigation, you explored the relationship between fractional exponents and square roots and important properties of radical expressions.a For n ≥ 0, what does √ $ n mean, and why does it make sense that √ $ n = n

1 _ 2 ?

b What property of square roots can be used to express √ $ n in equivalent, often simpler, forms?

c What formula gives the length of each diagonal in a square with sides of length s?

Be prepared to share your thinking with the entire class.